Test of Discrete Event Systems - 21.12.2020
Exercise 1
An electronic device driving the opening of a safe generates one of three numbers, 0, 1, or 2, according to the following rules:
i) if 0 was generated last, then the next number is 0 again with probability 1/2 or 1 with probability 1/2;
ii) if 1 was generated last, then the next number is 1 again with probability 2/5 or 2 with probability 3/5;
iii) if 2 was generated last, then the next number is either 0 with probability 7/10 or 1 with probability 3/10.
Moreover, the first generated number is 0 with probability 3/10, 1 with probability 3/10, and 2 with probability 2/5. The safe is opened the first time the sequence 120 takes place.
1. Define a discrete time Markov chain for the above described opening mechanism of the safe.
2. Compute the probability that the safe is opened with the fourth generated number.
3. Compute the average length of the sequence generated to open the safe.
Exercise 2
A study of the strengths of the basketball teams of the main American universities shows that, if a university has a strong team one year, it is equally likely to have a strong team or an average team next year; if it has an average team, it is average next year with 50% probability, and if it changes, it is just as likely to become strong as weak; if it is weak, it has 65% probability of remaining so, otherwise it becomes average.
1. What is the probability that a team is strong on the long run?
2. Assume that a team is strong. Compute the probability that it is strong for at least three consecutive years.
3. Assume that a team is weak. How many years are needed on average for it to become strong?
4. Assume that a team is weak. Compute the probability that, in the next ten years, it is strong at least two years.
